L(s) = 1 | + (1.41 − i)3-s + (−1.73 − 1.41i)5-s + (1.00 − 2.82i)9-s + 4.89·11-s − 4.89i·13-s + (−3.86 − 0.267i)15-s − 3.46·17-s + 3.46i·19-s + 6i·23-s + (0.999 + 4.89i)25-s + (−1.41 − 5.00i)27-s + 2.82i·29-s − 3.46i·31-s + (6.92 − 4.89i)33-s + 4.89i·37-s + ⋯ |
L(s) = 1 | + (0.816 − 0.577i)3-s + (−0.774 − 0.632i)5-s + (0.333 − 0.942i)9-s + 1.47·11-s − 1.35i·13-s + (−0.997 − 0.0691i)15-s − 0.840·17-s + 0.794i·19-s + 1.25i·23-s + (0.199 + 0.979i)25-s + (−0.272 − 0.962i)27-s + 0.525i·29-s − 0.622i·31-s + (1.20 − 0.852i)33-s + 0.805i·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.438 + 0.898i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.438 + 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.23604 - 0.771875i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.23604 - 0.771875i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.41 + i)T \) |
| 5 | \( 1 + (1.73 + 1.41i)T \) |
good | 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 - 4.89T + 11T^{2} \) |
| 13 | \( 1 + 4.89iT - 13T^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 19 | \( 1 - 3.46iT - 19T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 - 2.82iT - 29T^{2} \) |
| 31 | \( 1 + 3.46iT - 31T^{2} \) |
| 37 | \( 1 - 4.89iT - 37T^{2} \) |
| 41 | \( 1 - 5.65iT - 41T^{2} \) |
| 43 | \( 1 - 8.48T + 43T^{2} \) |
| 47 | \( 1 - 6iT - 47T^{2} \) |
| 53 | \( 1 - 10.3T + 53T^{2} \) |
| 59 | \( 1 - 4.89T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 8.48T + 67T^{2} \) |
| 71 | \( 1 + 9.79T + 71T^{2} \) |
| 73 | \( 1 + 9.79iT - 73T^{2} \) |
| 79 | \( 1 - 10.3iT - 79T^{2} \) |
| 83 | \( 1 - 6iT - 83T^{2} \) |
| 89 | \( 1 + 5.65iT - 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.11295210433868237090811133543, −11.26895759448820644195248030161, −9.745209479802425354812384773498, −8.880234610448207467519427796344, −8.056795974808688131978186849716, −7.21124132048154690843794399991, −5.93865566808242962550617917973, −4.27003326030152061369377699806, −3.26446269594474738312948334071, −1.30271004567595026115082456026,
2.35807344241981943403023715812, 3.87135830309401848052591851461, 4.46568894555772135039426875645, 6.55238397796224892640698454464, 7.26334025450709954761681223494, 8.718555189588517913444295952535, 9.117793333819184772937868637073, 10.42541761073981106867165832897, 11.30560895772376150069035814274, 12.08110284064545089628761713861